Developments in Mathematics
VOLUME 24
Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California
For further volumes: www.springer.com/series/5834 Jerzy Kakol ˛ Wiesław Kubis´ Manuel López-Pellicer
Descriptive Topology in Selected Topics of Functional Analysis Jerzy Kakol ˛ Wiesław Kubis´ Faculty of Mathematics and Informatics Institute of Mathematics A. Mickiewicz University Jan Kochanowski University 61-614 Poznan 25-406 Kielce Poland Poland [email protected] and Institute of Mathematics Manuel López-Pellicer Academy of Sciences of the Czech Republic IUMPA 115 67 Praha 1 Universitat Poltècnica de València Czech Republic 46022 Valencia [email protected] Spain and Royal Academy of Sciences 28004 Madrid Spain [email protected]
ISSN 1389-2177 ISBN 978-1-4614-0528-3 e-ISBN 978-1-4614-0529-0 DOI 10.1007/978-1-4614-0529-0 Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011936698
Mathematics Subject Classification (2010): 46-02, 54-02
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Springer is part of Springer Science+Business Media (www.springer.com) To our Friend and Teacher Prof. Dr. Manuel Valdivia Preface
We invoke (descriptive) topology recently applied to (functional) analysis of infinite-dimensional topological vector spaces, including Fréchet spaces, (LF)- spaces and their duals, Banach spaces C(X) over compact spaces X, and spaces Cp(X), Cc(X) of continuous real-valued functions on a completely regular Haus- dorff space X endowed with pointwise and compact–open topologies, respectively. The (LF)-spaces and duals particularly appear in many fields of functional analysis and its applications: distribution theory, differential equations and complex analysis, to name a few. Our material, much of it in book form for the first time, carries forward the rich legacy of Köthe’s Topologische lineare Räume (1960), Jarchow’s Locally Convex Spaces (1981), Valdivia’s Topics in Locally Convex Spaces (1982), and Pérez Car- reras and Bonet’s Barrelled Locally Convex Spaces (1987). We assume their (stan- dard English) terminology. A topological vector space (tvs) must be Hausdorff and have a real or complex scalar field. A locally convex space (lcs) is a tvs that is lo- cally convex. Engelking’s General Topology (1989) serves as a default reference for general topology. The authors wish to thank Professor B. Cascales, Professor M. Fabian, Professor V. Montesinos, and Professor S. Saxon for their valuable comments and suggestions, which made this material much more readable. The research of J. Kakol ˛ was partially supported by the Ministry of Science and Higher Education, Poland, under grant no. NN201 2740 33. W. Kubis´ was supported in part by grant IAA 100 190 901, by the Institutional Research Plan of the Academy of Sciences of the Czech Republic under grant no. AVOZ 101 905 03, and by an internal research grant from Jan Kochanowski Uni- versity in Kielce, Poland. The research of J. Kakol ˛ and M. López-Pellicer was partially supported by the Spanish Ministry of Science and Innovation, under project no. MTM 2008-01502. Poznan, Poland Jerzy Kakol ˛ Kielce, Poland Wiesław Kubis´ Valencia, Spain Manuel López-Pellicer
vii Contents
1 Overview ...... 1 1.1 General comments and historical facts ...... 7 2 Elementary Facts about Baire and Baire-Type Spaces ...... 13 2.1 Baire spaces and Polish spaces ...... 13 2.2 A characterization of Baire topological vector spaces ...... 18 2.3 Arias de Reyna–Valdivia–Saxon theorem ...... 20 2.4 Locally convex spaces with some Baire-type conditions ...... 24 2.5 Strongly realcompact spaces X and spaces Cc(X) ...... 36 2.6 Pseudocompact spaces, Warner boundedness and spaces Cc(X) .. 46 2.7 Sequential conditions for locally convex Baire-type spaces ..... 56 3 K-analytic and Quasi-Suslin Spaces ...... 63 3.1Elementaryfacts...... 63 3.2 Resolutions and K-analyticity ...... 71 3.3 Quasi-(LB)-spaces ...... 82 3.4 Suslin schemes ...... 91 3.5 Applications of Suslin schemes to separable metrizable spaces . . . 93 3.6 Calbrix–Hurewicz theorem ...... 101 4 Web-Compact Spaces and Angelic Theorems ...... 109 4.1 Angelic lemma and angelicity ...... 109 4.2 Orihuela’s angelic theorem ...... 111 4.3 Web-compact spaces ...... 113 4.4 Subspaces of web-compact spaces ...... 116 4.5 Angelic duals of spaces C(X) ...... 118 4.6 About compactness via distances to function spaces C(K) .....120 5 Strongly Web-Compact Spaces and a Closed Graph Theorem ....137 5.1 Strongly web-compact spaces ...... 137 5.2 Products of strongly web-compact spaces ...... 138 5.3 A closed graph theorem for strongly web-compact spaces .....140
ix x Contents
6 Weakly Analytic Spaces ...... 143 6.1 A few facts about analytic spaces ...... 143 6.2 Christensen’s theorem ...... 149 6.3 Subspaces of analytic spaces ...... 155 6.4 Trans-separable topological spaces ...... 157 6.5 Weakly analytic spaces need not be analytic ...... 164 6.6 More about analytic locally convex spaces ...... 167 6.7 Weakly compact density condition ...... 168 6.8 More examples of nonseparable weakly analytic tvs ...... 174 7 K-analytic Baire Spaces ...... 183 7.1 Baire tvs with a bounded resolution ...... 183 7.2 Continuous maps on spaces with resolutions ...... 187 8 A Three-Space Property for Analytic Spaces ...... 193 8.1AnexampleofCorson...... 193 8.2 A positive result and a counterexample ...... 196
9 K-analytic and Analytic Spaces Cp(X) ...... 201 9.1 A theorem of Talagrand for spaces Cp(X) ...... 201 9.2 Theorems of Christensen and Calbrix for Cp(X) ...... 204 9.3 Bounded resolutions for Cp(X) ...... 215 9.4 Some examples of K-analytic spaces Cp(X) and Cp(X, E) ....230 9.5 K-analytic spaces Cp(X) over a locally compact group X .....231 ∧ 9.6 K-analytic group Xp ofhomomorphisms...... 234 10 Precompact Sets in (LM)-Spaces and Dual Metric Spaces ...... 239 10.1 The case of (LM)-spaces: elementary approach ...... 239 10.2 The case of dual metric spaces: elementary approach ...... 241 11 Metrizability of Compact Sets in the Class G ...... 243 11.1 The class G:examples...... 243 11.2 Cascales–Orihuela theorem and applications ...... 245 12 Weakly Realcompact Locally Convex Spaces ...... 251 12.1 Tightness and quasi-Suslin weak duals ...... 251 12.2 A Kaplansky-type theorem about tightness ...... 254 12.3 K-analytic spaces in the class G ...... 258 12.4 Every WCG Fréchet space is weakly K-analytic ...... 260 12.5 Amir–Lindenstrauss theorem ...... 266 12.6AnexampleofPol...... 271 12.7 More about Banach spaces C(X) over compact scattered X ....276 13 Corson’s Property (C) and Tightness ...... 279 13.1 Property (C) and weakly Lindelöf Banach spaces ...... 279 13.2 The property (C) for Banach spaces C(X) ...... 284 Contents xi
14 Fréchet–Urysohn Spaces and Groups ...... 289 14.1 Fréchet–Urysohn topological spaces ...... 289 14.2 A few facts about Fréchet–Urysohn topological groups ...... 291 14.3 Sequentially complete Fréchet–Urysohn spaces are Baire .....296 14.4 Three-space property for Fréchet–Urysohn spaces ...... 299 14.5 Topological vector spaces with bounded tightness ...... 302 15 Sequential Properties in the Class G ...... 305 15.1 Fréchet–Urysohn spaces are metrizable in the class G ...... 305 15.2 Sequential (LM)-spaces and the dual metric spaces ...... 311 − 15.3 (LF )-spaces with the property C3 ...... 320 16 Tightness and Distinguished Fréchet Spaces ...... 327 16.1 A characterization of distinguished spaces ...... 327 16.2 G-basesandtightness...... 334 16.3 G-bases, bounding, dominating cardinals, and tightness .....338 16.4 More about the Wulbert–Morris space Cc(ω1) ...... 349 17 Banach Spaces with Many Projections ...... 355 17.1 Preliminaries, model-theoretic tools ...... 355 17.2 Projections from elementary submodels ...... 361 17.3 Lindelöf property of weak topologies ...... 364 17.4 Separable complementation property ...... 365 17.5 Projectional skeletons ...... 369 17.6 Norming subspaces induced by a projectional skeleton ...... 375 17.7 Sigma-products ...... 380 17.8 Markushevich bases, Plichko spaces and Plichko pairs ...... 383 17.9 Preservation of Plichko spaces ...... 388 18 Spaces of Continuous Functions over Compact Lines ...... 395 18.1 General facts ...... 395 18.2 Nakhmanson’s theorem ...... 398 18.3 Separable complementation ...... 399 19 Compact Spaces Generated by Retractions ...... 405 19.1Retractiveinversesystems...... 405 19.2 Monolithic sets ...... 409 19.3 Classes R and RC ...... 411 19.4 Stability ...... 412 19.5Someexamples...... 415 19.6 The first cohomology functor ...... 418 19.7 Compact lines ...... 422 19.8 Valdivia and Corson compact spaces ...... 425 19.9 Preservation theorem ...... 432 19.10 Retractional skeletons ...... 434 19.11 Primarily Lindelöf spaces ...... 438 19.12 Corson compact spaces and WLD spaces ...... 440 19.13 A dichotomy ...... 442 xii Contents
19.14 Alexandrov duplications ...... 446 19.15 Valdivia compact groups ...... 448 19.16 Compact lines in class R ...... 451 19.17 More on Eberlein compact spaces ...... 456 20 Complementably Universal Banach Spaces ...... 467 20.1Amalgamationlemma...... 467 20.2 Embedding-projection pairs ...... 469 20.3 A complementably universal Banach space ...... 471 References ...... 475 Index ...... 491 Chapter 1 Overview
Let us briefly describe the organization of the book. Chapter 2, essential to the sequel, contains classical results about Baire-type con- ditions (Baire-like, b-Baire-like, CS-barrelled, s-barrelled) on tvs. We include appli- cations to closed graph theorems and C(X) spaces. We also provide the first proof in book form of a remarkable result of Saxon [355] (extending earlier results of Arias de Reyna and Valdivia) that states that, under Martin’s axiom, every lcs containing a dense hyperplane contains a dense non-Baire hyperplane. Ours, then, is the first book to solve the first problem formally posed in Pérez Carreras and Bonet’s excel- lent monograph. Chapter 2 also contains analytic characterizations of certain com- pletely regular Hausdorff spaces X. For example, we show that X is pseudocom- pact, is Warner bounded, or Cc(X) is a (df )-space if and only if for each sequence (μn)n in the dual Cc(X) there exists a sequence (tn)n ⊂ (0, 1] such that (tnμn)n is weakly bounded, strongly bounded, or equicontinuous, respectively ([231, 232]). These characterizations help us produce a (df )-space Cc(X) that is not a (DF )- space [232], solving a basic and long-standing open question. The third characteri- zation is joined by nine more that supply tenfold an implied Jarchow request. These forge a strong link we happily claim between his book and ours. Chapter 3 deals with the K-analyticity of a topological space E and the con- { : ∈ NN} cept of a resolution generated on E (i.e., a family of sets Kα α such = ⊂ ≤ that E α Kα and Kα Kβ if α β). Compact resolutions (i.e., resolutions N {Kα : α ∈ N } whose members are compact sets) naturally appear in many situa- tions in topology and functional analysis. Any K-analytic space admits a compact resolution [388], and for many topological spaces X the existence of such a resolu- tion is enough for X to be K-analytic; (see [80], [82]). Many of the ideas in the book are related to the concept of compact resolution and are already in or have been in- spired by papers [388], [80], [82]. It is an easy and elementary exercise to observe that any separable metric and complete space E admits a compact resolution, even swallowing compact sets. In Chapter 3, we gather some results, mostly due to Val- divia [421], about lcs (called quasi-(LB)-spaces) admitting resolutions consisting of Banach discs and their relations with the closed graph theorems.
J. Kakol ˛ et al., Descriptive Topology in Selected Topics of Functional Analysis, 1 Developments in Mathematics 24, DOI 10.1007/978-1-4614-0529-0_1, © Springer Science+Business Media, LLC 2011 2 1Overview
These concepts are related to another one, called a Suslin scheme, which pro- vides a powerful tool to study structural properties of metric separable spaces (see [175], [346]). Chapter 3 presents Hurewicz and Alexandrov’s theorems as well as the Calbrix–Hurewicz theorem, which yields that a regular analytic space X (i.e., a continuous image of the space NN)isnotσ -compact if and only if X contains a closed subset homeomorphic to NN. We have tried to present proofs in a transparent form. The reader is also referred to the magnificent works [387], [388], [421], [346], [99], among others. Chapter 4 deals with the class of angelic spaces, introduced by Fremlin, for which several variants of compactness coincide. A remarkable paper of Orihuela [320]in- troduces a large class of topological spaces X (under the name web-compact)for which the space Cp(X) is angelic. Orihuela’s theorem covers many already known partial results providing Eberlein–Šmulian-type results. Following Orihuela [320], we show that Cp(X) is angelic if X is web-compact. This yields, in particular, Ta- lagrand’s result [388] stating that for a compact space X the space Cp(X) is K- analytic if and only if C(X) is weakly K-analytic. In Chapter 4, we present some quantitative versions of Grothendieck’s characterization of the weak compactness for spaces C(X) (for compact Hausdorff spaces X) and quantitative versions of the classical Eberlein–Grothendieck and Krein–Šmulian theorems. We follow very re- cent works of Angosto and Cascales [6], [7], [10], Angosto [9], Angosto, Cascales and Namioka [8], Cascales, Marciszewski and Raja [92], Hájek, Montesinos and Zizler [150] and Granero [187]. The last two articles, [150] and [187], where in the case of Banach spaces these quantitative generalizations have been studied and presented, motivated the other papers mentioned. In Chapter 5, we continue the study of web-compact spaces. A subclass of web- compact spaces, called strongly web-compact, is introduced, and a closed graph theorem for such spaces is provided. We prove that an own product of a strongly web-compact space need not be web-compact. This shows that there exists a quasi- Suslin space X such that X × X is not quasi-Suslin. Chapter 6 studies analytic spaces. We show that a regular space X is analytic if and only if X has a compact resolution and admits a weaker metric topology. This fact, essentially due to Talagrand [392], extended Choquet’s theorem [97](ev- ery metric K-analytic space is analytic); see also [85]. Several applications will be provided. We show Christensen’s theorem [99] stating that a separable metric topo- logical space X is a Polish space if and only if X admits a compact resolution swal- lowing compact sets. The concept of a compact resolution swallowing compact sets is already present in the main result of [82, Theorem 1]. We study trans-separable spaces and show that a tvs with a resolution of precompact sets is trans-separable [342]. This serves to prove [82] that precompact sets are metrizable in any uniform space whose uniformity admits a U -basis. Consequences are provided. Chapter 6 also works with the following general problem (among some others): When can analyticity or K-analyticity of the weak topology σ(E,E ) of a dual pair (E, E ) be lifted to stronger topologies on E compatible with the dual pair? The question is essential since (as we show) there exist many weakly analytic lcs’s (i.e., analytic in the weak topology σ(E,E ) that are not analytic. We prove that if X is 1Overview 3 an uncountable analytic space, the Mackey dual Lμ(X) of Cp(X) is weakly analytic and not analytic. The density condition due to Heinrich [203], studied in a series of papers of Bierstedt and Bonet [51], [52], [53], [54], [55], motivates a part of Chapter 6 that studies the analyticity of the Mackey and strong duals of (LF )-spaces. In Chapter 7, we show that a tvs that is a Baire space and admits a countably compact resolution is metrizable, separable and complete. This extends a classical result of De Wilde and Sunyach [111] and Valdivia’s theorem [421]. An interesting recent applicable result due to Drewnowski (highly inspired by [233]) about con- tinuous maps between F-spaces is presented. Namely, we show that a linear map N T : E → F from an F-space E having a resolution {Kα : α ∈ N } into a tvs F is continuous if each restriction T |Kα is continuous. This theorem was motivated by the Arias–De Reina–Valdivia–Saxon theorem about non-Baire dense hyperplanes in Banach spaces. We provide a large class of weakly analytic metrizable and separable Baire tvs’s not analytic (clearly such spaces are necessarily not locally convex). Examples of spaces of this type will be used in Chapter 8 to prove that analyticity is not a three-space property. We prove, however, that a metrizable topological vec- tor space E is analytic if it contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. We reprove also in Chapter 8 (using Corson’s example [103]) that the Lindelöf property is not a three-space property. Chapter 9 partially continues the study started in Chapter 3 and deals with K- analytic and analytic spaces Cp(X). Some results due to Talagrand [388], Tkachuk [399], Velichko [27] and Canela [78] are presented. We extend the main result of [399] characterizing K-analytic spaces Cp(X) in terms of resolutions. Christensen’s remarkable theorem [99] stating that a metrizable and separable space X is σ - compact if and only if Cp(X) is analytic is proved. We show that the analyticity of Cp(X) for any X implies that X is σ -compact, from Calbrix [77]. A charac- terization of σ -compactness of a cosmic space X in terms of subspaces of RX is provided from Arkhangel’skii and Calbrix [30]. Finally, we show that Cp(X) is K- X analytic-framed in R if and only if Cp(X) admits a bounded resolution [160]. We also collect several equivalent conditions for a space Cp(X) to be a Lindelöf space over locally compact groups X;see[234]. Chapter 10, which might be a good motivation for Chapters 11 and 12, extends the main result of Cascales and Orihuela [81] and presents the unified and direct proofs [229]ofPfister[329], Cascales and Orihuela [81] and Valdivia’s [423] theo- rems about the metrizability of precompact sets in (LF )-spaces, (DF )-spaces and dual metric spaces, respectively. The proofs from [229] do not require the typical machinery of quasi-Suslin spaces, upper semicontinuous compact-valued maps, and so on. Chapter 11 introduces (after Cascales and Orihuela [83]) a large class of locally convex spaces under the name the class G.AnlcsE is said to be in the class G if its N topological dual E admits a resolution {Aα : α ∈ N } such that sequences in each Aα are equicontinuous. The class G contains among others all (LM)-spaces (hence (LF )-spaces) and dual metric spaces (hence (DF )-spaces), spaces of distributions D (Ω), spaces A(Ω) of real analytic functions on open Ω ⊂ Rn and others. We 4 1Overview show in Chapter 11 the main result of [83], with a simpler proof from [155], stating that every precompact set in an lcs in the class G is metrizable. This general re- sult covers many already known theorems for (DF )-spaces, (LF )-spaces and dual metric spaces. In Chapter 12, we continue the study of spaces in the class G. We prove that the weak∗ dual (E ,σ(E ,E)) of an lcs E in the class G is K-analytic if and only if (E ,σ(E ,E)) is Lindelöf if and only if (E, σ (E, E )) has countable tightness if and only if each finite product (E ,σ(E ,E))n is Lindelöf; see [88]. Develop- ing the argument producing upper semicontinuous maps, we also show that every quasibarrelled space in the class G has countable tightness both for the weak and the original topologies. This extends a classical result of Kaplansky for a metrizable lcs; see [165]. Although (DF )-spaces belong to the class G, concrete examples of (DF )-spaces without countable tightness are provided. On the other hand, there are many Banach spaces E for which E endowed with the weak topology σ(E,E ) is not Lindelöf. We show, however, from Khurana [242], that every weakly compactly generated (WCG) Fréchet space E is weakly K-analytic (i.e., (E, σ (E, E )) is K-analytic). This extends Talagrand’s correspond- ing result for WCG Banach spaces; see [390] and also [149], [323]. In general, for a WCG lcs, this result fails, as Hunter and Lloyd [209] have shown. It is natural to ask (Corson [103];seealso[271]) if every weakly Lindelöf Banach space is a WCG Banach space (i.e., if E admits a weakly compact set whose linear span is dense in E). The first example of a non-WCG Banach space whose weak topology is Lindelöf was provided by Rosenthal [349]. We present an example due to Pol [336]show- ing that there exists a Banach space C(X) over a compact scattered space X such that C(X) is weakly Lindelöf and C(X) is not a WCG Banach space. This example also answers (in the negative) some questions of Corson [103, Problem 7], posed by Benyamini, Rudin and Wage from [49]. Talagrand, inspired and motivated by sev- eral results of Corson, Lindenstrauss and Amir, continued this line of research in his remarkable papers (see, e.g., [388], [389], [390], [391]). Chapter 12 also contains the proof of the Amir–Lindenstrauss theorem that every nonseparable reflexive Banach space contains a complemented separable subspace [270]. Several consequences are provided. This subject, related to WCG Banach spaces and the Amir–Lindenstrauss theorem, will be continued in Chapters 17, 18, 19 and 20, where Banach spaces with a rich family of projections onto separable subspaces are studied. In Chapter 13, the class of Banach spaces having the property (C) is studied. This property, isolated by Corson [103], provides a large subclass of Banach spaces E whose weak topology need not be Lindelöf. We collect some results of Corson [103], Pol [334], [338] and Frankiewicz, Plebanek and Ryll-Nardzewski [168]. Chapters 14 and 15 deal with topological (vector) spaces satisfying some sequen- tial conditions. We study Fréchet–Urysohn spaces (i.e., spaces E such that for each A ⊂ E and each x ∈ A there exists a sequence in A converging to x). The main result states that every sequentially complete Fréchet–Urysohn lcs is a Baire space. Since every infinite-dimensional Montel (DF )-space E is nonmetrizable and sequential (i.e., every sequentially closed set in E is closed), the following question arises: Is every Fréchet–Urysohn space in the class G metrizable? 1Overview 5
In Chapter 15, we prove that an lcs in the class G is metrizable if and only if E is b-Baire-like and if and only if E is Fréchet–Urysohn; see [88], [89]. Consequently, no proper (LB)-space E is Fréchet–Urysohn (since E contains the space ϕ; i.e., the ℵ0-dimensional vector space with the finest locally convex topology). We prove that if a (DF )-or(LM)-space E is sequential, then E is either metrizable or Montel (DF );see[229]. Webb [415] introduced the property C3 (i.e., sequential closure of any set is sequentially closed), which characterizes metrizability for (LM)-spaces but not for (DF )-spaces. We distinguish a variant of the property C3 called prop- − erty C3 (i.e., sequential closure of any vector subspace is sequentially closed) and − characterize both (DF )-spaces and (LF )-spaces with the property C3 as being of the form M, φ,orM × φ, where M is metrizable [229]. In Chapter 16, we apply the concept of tightness to study distinguished Fréchet spaces. Valdivia provided a nondistinguished Fréchet space whose weak∗ bidual is quasi-Suslin but not K-analytic; see [421]. Using the concept of tightness, we show that Köthe’s echelon nondistinguished Fréchet space λ1(A) serves the same purpose [157], and we provide another (much simpler) proof of the deep result of Bastin and Bonet stating that for λ1(A) there exists a locally bounded discontinu- ous linear functional over the space (λ1,β(λ1,λ1));see[158]. The basic fact [157] is that a (DF )-space is quasibarrelled if and only if its tightness is countable. We show that a Fréchet space is distinguished if and only if its strong dual has count- able tightness. This approach to studying distinguished Fréchet spaces leads to a rich supply of (DF )-spaces whose weak∗ duals are quasi-Suslin but not K-analytic, for example spaces Cc(κ) for κ a cardinal of uncountable cofinality. The small car- dinals b and d will be used to improve the analysis of Köthe’s example; see [159], [157]. The bounding cardinal b (introduced by Rothberger) is the smallest infinite- dimensionality for metrizable barrelled spaces; see [357] for details. In general, a quasibarrelled space E belongs to the class G if and only if E N admits a G-basis (i.e., a family {Uα : α ∈ N } of neighborhoods of zero in E such that every neighborhood of zero in E contains some Uα;see[157]). This concept provides spaces Cc(X) different from those Talagrand presented in [388], whose weak∗ dual is not K-analytic but does have a compact resolution. We show that the weak∗ dual of any space in the class G is quasi-Suslin [159]. An immediate consequence is that every space with a G-basis enjoys this property. In Chapter 16, we show that Cc(ω1) may or may not have a G-basis. The existence of a G-basis for Cc(ω1) depends on the axioms of set theory. Cc(ω1) has a G-basis if and only if ℵ1 = b. Several interesting examples of (DF )-spaces that admit and do not admit G-bases will also be provided. In Chapter 17, we continue the subject developed in Chapter 12 related to WCG Banach spaces and the Amir–Lindenstrauss theorem. We discuss Banach spaces that have a rich family of norm-one projections onto separable subspaces. Probably, the most general class of Banach spaces with “many” projections specifies the sepa- rable complementation property. Recall that a Banach space E has the separable complementation property (SCP) if for every separable subspace D of E there ex- ists a bounded linear projection with a separable range containing D. This property seems to be too general for proving any reasonable structural properties of Banach 6 1Overview spaces. A strengthening of the SCP is the notion of a projectional skeleton, defined and studied in Section 17.5. Banach spaces with a projectional skeleton have good stability properties as well as some nice structural ones. For instance, they have an equivalent locally uniformly convex norm and admit a bounded injective linear operator into some c0(Γ )-space. A natural property of a projectional skeleton is commutativity. It turns out that this is equivalent to the existence of a countably norming Markushevich basis. A space with this property is often called a Plichko space. We study this class of spaces in Sect. 17.7. A more special property of a projectional skeleton gives the class of WLD Banach spaces. We present selected results concerning the com- plementation property in general Banach spaces in order to motivate the study of projectional skeletons and projectional resolutions. This section contains some in- formation about Plichko spaces, stability of this class and some natural examples. In order to simplify several arguments and constructions of projections, we present in Section 17.1 the method of elementary substructures coming from logic. Using this method we prove, for example, a result on the Lindelöf property of the topology induced by a certain norming subspace of the dual and by the projectional skeleton. In the case of WLD spaces, this is the well-known result on the Lindelöf property of the weak topology. The method of elementary substructures is also used in the following chapters for proving topological properties such as countable tight- ness. Chapter 18 discusses selected properties of Banach spaces of type C(X), where X is a linearly ordered compact space, called a compact line for short. In particular, we present Nakhmanson’s theorem stating that if X is a compact line such that Cp(X) is a Lindelöf space, then X is second-countable. Compact lines are relatively easy to investigate, yet they form a rich class of spaces and provide several interesting examples. A very special case is the smallest uncountable well-ordered space, ω1 +1, which appears several times in the previous chapters. Its space of continuous functions turns out to be a canonical example for several topological and geometric properties of Banach spaces. More complicated compact lines provide examples related to Plichko spaces. Chapter 19 presents several classes of nonmetrizable compact spaces that corre- spond to well-known classes of Banach spaces with many projections. In particular, we discuss the class of Valdivia compact spaces and its subclasses: Corson and Eberlein compact spaces. We discuss a general class of compact spaces obtained by limits of continuous retractive sequences. We also introduce the notion of a retrac- tional skeleton, dual to projectional skeletons in Banach spaces. The last section of Chapter 19 contains an overview of Eberlein compact spaces, with some classical results and examples relevant to the subject of previous chapters. Finally, Chapter 20 deals with complementably universal Banach spaces. As- suming the continuum hypothesis, there exists a complementably universal Banach space of density ℵ1 for the class of Banach spaces with a projectional resolution of the identity. Similar methods produce a universal preimage for the class of Valdivia compact spaces of weight ℵ1. 1.1 General comments and historical facts 7
1.1 General comments and historical facts
The earliest approach to K-analytic spaces seems to be due to Choquet [96], who called a topological space K-analytic if it is a Kσδ-set in some compact space. Rogers proved [345] that if X is a completely regular Hausdorff space, the last condition is equivalent to the following one: (*) There exists a Polish space Y and an upper semicontinuous compact-valued map from Y covering X. The property (*) can be seen in Martineau [282]; see also Frolik [177], Rogers [345], Stegall [387] and Sion [380], [381]. In our book, by a K-analytic space X we mean a topological space satisfying condition (*). Since every Polish space is a continuous image of the space NN, we note another equivalent way to look at K-analytic spaces is as the image under an upper semicontinuous compact-valued map of the space NN. The images under upper semicontinuous compact set-valued maps of Polish spaces are also known in the literature as K-Suslin spaces; see also [421], [282], [177], [345]. K-analytic and analytic spaces are also useful topological objects for the study of nice properties of topological measures; see [371], [170]. For example, every semifinite topological measure that is inner regular for closed sets for a K-analytic space is inner regular for compact sets, and every semifinite Borel measure is inner regular for compact sets for an analytic space. Every K-analytic N space X admits a compact resolution (i.e., a family {Kα : α ∈ N } of compact sets covering X such that Kα ⊂ Kβ if α ≤ β); see [388], [80], [82], [131]—(in [131] this term was used formally for the first time). In the frame of angelic spaces X,the existence of a compact resolution implies that X is necessarily a K-analytic space [80]. Talagrand [388] had already observed this for spaces Cp(X) over compact spaces X; see also [85]. Many interesting topological problems in infinite-dimensional topological vector spaces might be motivated by some results from the theory of Cp(X) spaces. Let us mention, for example, one of them due to Velichko [27, Theorem I.2.1]: The space Cp(X) is σ -compact (i.e., covered by a sequence of compact sets) if and only if X is finite. Tkachuk and Shakhmatov [403] extended this result to σ -countably compact spaces Cp(X); see also [25] for a general approach including both cases. Clearly, if (Kn)n is an increasing sequence of compact sets covering Cp(X), then the sets := = ∈ NN Kα Kn1 , where α (nk) , form a compact resolution for Cp(X).Onthe other hand, we can prove that Cp(X) has a fundamental sequence of bounded sets only if X is finite, see Chapter 2. Natural questions arise: Characterize completely regular Hausdorff spaces X for which Cp(X) admits a compact resolution. When does Cp(X) admit a resolution consisting of topologically bounded sets? Recently, a related problem has been solved by Tkachuk [399], who proved: Cp(X) is K-analytic if and only if Cp(X) admits a compact resolution. We provide another approach to solving this problem. We show that if Cp(X) admits a resolution consisting of tvs-bounded sets (i.e., sets absorbed by any neigh- 8 1Overview borhood of zero in Cp(X)) then Cp(X) is angelic. Since angelic spaces with com- pact resolutions are K-analytic [80], this yields Tkachuk’s result and provides more applications. The class of weakly Lindelöf determined Banach spaces (WLD Banach spaces, introduced in [13]) provides a larger class of weakly Lindelöf Banach spaces con- taining the weakly compactly generated (WCG) Banach spaces. The study of WLD Banach spaces was motivated by results of Gul’ko [196] about weakly K-countably determined Banach spaces (also called weakly countably determined WCD Banach spaces, weakly Lindelöf Σ-spaces or Vašák spaces ([388], [418];seealso[147], [298]). Recall that according to [322] a Banach space E is WLD if and only if its closed unit ball in E is Corson compact in σ(E ,E). Quite recently, Cascales, Namioka and Orihuela [91] have shown that if E is a Banach space satisfying Cor- son’s property (C) and E admits a projectional generator, then E is WLD. We re- prove this result in Section 17.6 using the notion of a projectional skeleton. See also [84], [149] (and references therein) for more details. In 1961, H. Corson [103] started a systematic study of certain topological prop- erties of the weak topology of Banach spaces. This line of research provided more general classes such as reflexive Banach spaces, weakly compactly generated Ba- nach spaces [5], [349], [121] and the class of weakly K-analytic and weakly K- countably determined Banach spaces. For another approach to studying geometric and topological properties of nonseparable Banach spaces, we refer to [200]. In his fundamental paper [103], Corson asked if WCG Banach spaces are exactly those Banach spaces whose weak topology is Lindelöf. The first example of a non- WCG Banach space whose weak topology was Lindelöf was provided by Rosenthal [349]. One can ask for which compact spaces X the space Cp(X) is Lindelöf. This problem was first studied by Corson [103];seealso[105]. The class of Corson compact spaces X (i.e., homeomorphic to a compact subset of a Σ-product of real lines) provides examples of Lindelöf spaces Cp(X);see[4], [195]. On the other hand, for every weakly K-analytic Banach space E, the closed unit ball in E in the topology σ(E ,E) is a Corson compact set [196]. There exist, however, examples due to Talagrand, Haydon and Kunen (under the continuum hypothesis (CH)) of Corson compact spaces X such that the Banach space C(X) is not weakly Lindelöf; see [312]. Also, in [15] it was shown that for a Corson compact space X the Banach space C(X) is WLD if and only if every positive regular Borel measure on X has separable support. In general, the claim that for every Corson compact space X the space C(X) is a WLD Banach space is independent of the usual axioms of set theory [15]. There exist concrete Banach spaces C(X) over compact scattered spaces X that are weakly Lindelöf but not WCG; see, for example, [338], [388]. Nevertheless, for a compact space X, the Banach space C(X) is WCG if and only if X is Eberlein compact [5]; see also [147], [149]. For a compact space X, the space C(X) is weakly K-analytic if and only if Cp(X) is K-analytic [388]. This distinguishes the class of Talagrand compact spaces (i.e., compact X for which Cp(X) is K-analytic). This line of research between topology and functional analysis inspired several specialists (mainly from functional analysis) to develop new techniques from de- 1.1 General comments and historical facts 9 scriptive topology to study concrete problems and classes of spaces in linear func- tional analysis; see [149] for many references. For example, one may ask: (i) Is a Banach space E weakly Lindelöf if its weak∗ dual (E ,σ(E ,E)) has countable tightness? (ii) If σ(E,E ) is Lindelöf, is the unit ball in E of countable tightness in σ(E ,E)? Question (ii) is related to Banach spaces satisfying the property (C) of Corson. This property, introduced by Corson [103], provided a large subclass of Banach spaces E whose weak topology need not be Lindelöf. Papers of Corson and Pol described the property (C) in terms of countable tightness-type conditions for the topology σ(E ,E). Corson’s paper [103] concerning the property (C) and results of Pol from [334], [336]or[337] motivated several articles, for example [323], [331], [332], [168], [88], among others, to study concrete classes of the weakly Lindelöf Banach spaces. This subject of research has been continued by many specialists; we refer the reader to articles [80], [83], [78], [293][294];seealso[88], [89], [85], [159], [157], [158], [131]. Many important spaces in functional analysis are defined as certain (DF )- spaces, (LB)-spaces, or (LF )-spaces (i.e., inductive limits of a sequence of Banach (Fréchet) spaces), or their strong duals; see, for example, [51], [52], [54], [288], [328], [213], [421] as good sources of information. A significant difference from the Banach space case is that the strong dual of a Fréchet space is not metrizable in general. The strong duals of Fréchet spaces are (DF )-spaces, introduced by Grothendieck [188]. Clearly, any (LB)-space is a (DF )-space. One can ask, among other things, for which (LF )-spaces E their Mackey dual (E ,μ(E ,E)) or strong dual (E ,β(E ,E)) is K-analytic or even analytic. It was known already [83] that the precompact dual of any separable (LF )- space is analytic. In [367], [368], the class of strongly weakly compactly generated (SWCG) Banach spaces E for which the Mackey topology μ(E ,E) arises in a natural way, was introduced. There are many interesting topological problems related to the classes of lcs’s above. Let us mention a few of them strictly connected with the topic of the book. Floret [166], motivated by earlier works of Grothendieck, Fremlin, De Wilde and Pryce, asked if the compact sets in any (LF )-space are metrizable and if any (LF )- space is weakly angelic. Although the first question (as we have already mentioned in the Preface) has been answered positively for (DF )-spaces and dual metric spaces, both questions for (LF )-spaces were solved (also positively) by Cascales and Orihuela in [81]. Orihuela [320] answered the second question also for dual metric spaces. Therefore, it was natural to ask about a possible large class of lcs (clearly including (LF )-spaces and dual metric spaces) for which both questions also have positive answers. Such a class of lcs, called the class G, was introduced by Cascales and Orihuela [82]. An lcs E belongs to G if the weak∗ dual E admits a N resolution {Kα : α ∈ N } consisting of σ(E ,E) relatively countably compact sets such that each sequence in any Kα is equicontinuous. Spaces in the class G enjoy interesting topological properties, such as: 10 1Overview
(i) The weak topology of E ∈ G is angelic, and every precompact set in E is metrizable. (ii) For E ∈ G, the densities of E and (E ,σ(E ,E)) coincide if (E, σ (E, E )) is a Lindelöf Σ-space [82]; this extends a classical result of Amir and Lindenstrauss for WCG Banach spaces. (iii) For a compact space X, the space Cp(X) is K-analytic if and only if it is homeomorphic to a weakly compact set of a locally convex space in the class G, see [82]. Recall that a compact space X is Eberlein compact if and only if it is homeomorphic to a weakly compact subset of a Banach space. (iv) An lcs in G is metrizable if and only if it is Fréchet–Urysohn. A barrelled lcs E in the class G (for example, any (LF )-space E) is metrizable if and only if E is Fréchet–Urysohn, if and only if E is Baire-like if and only if E does not contain φ (i.e., the ℵ0-dimensional vector space with the finest locally convex topology [89]). (v) Every quasibarrelled space E in G has countable tightness, and the same also holds true for (E, σ (E, E )) [88]. This extends a classical results of Kaplansky; see [165]. On the other hand, there is a large and important class of lcs that does not belong to G.AnlcsCp(X) belongs to the class G if and only if X is countable (i.e., Cp(X) is metrizable); see [89]. Therefore, many results for spaces Cp(X) (also presented in the book) require methods and techniques different from those applied to study the class G; we refer the reader to the excellent works about Cp(X) theory in [25] and [24]. Recall that a topological space X is sequential if every sequentially closed set in X is closed. Clearly, from the definition, we have metrizable ⇒ Fréchet–Urysohn ⇒ sequential ⇒ k-space. Probably the first proof that an (LF )-space is metrizable if and only if it is Fréchet–Urysohn was presented in [224]. Cascales and Orihuela’s [81] result that (LF )-spaces are angelic proved that any (LF )-space is sequential if and only if it is a k-space. Nyikos [316] observed that the (LB)-space φ is sequential and not Fréchet– Urysohn. Much earlier, Yoshinaga [436] had proved that the strong dual of any Fréchet–Schwartz space (equivalently, every Silva space) is sequential. Next, Webb [415] extended this result to strong duals of Fréchet–Montel spaces (equivalently, to Montel-(DF )-spaces). He proved that these spaces are Fréchet–Urysohn only when finite-dimensional. Since Montel (DF )-spaces form a part of the class of (LB)-spaces, one can ask if the Nyikos–Yoshinaga–Webb result extends further within (LB)-spaces or (DF )- spaces. It turns out [229] that the answers are negative. The strong dual of a metrizable lcs E is sequential if and only if E is a dense subspace of either a Banach space or a FréchetÐMontel space. Apparently, any proper (LB)-space is not Fréchet–Urysohn since proper (LB)- spaces contain a copy of φ;see[356]. As we have already mentioned, Fréchet– Urysohn (DF )-spaces or (LF )-spaces are metrizable since they belong to the class G. At this point, there emerges a disparity. Namely, Webb [415] introduced the property C3 (sequential closure of each set is sequentially closed), which charac- terizes metrizability for (LF )-spaces; [224] but not for (DF )-spaces; see [229,As- − sertions 5.2 and 5.3]. A variant property C3 (the sequential closure of every linear 1.1 General comments and historical facts 11 subspace is sequentially closed), defined in [229], characterized both the barrelled (DF )-spaces and (LF )-spaces as being of the form M, φ,orM × φ, where M is a metrizable lcs. The main result of [229], characterizing both (DF )-spaces and (LF )-spaces that are sequential as being either metrizable or Montel (DF )-spaces, provides an answer to topological group questions of Nyikos [316, Problem 1]. Chapter 2 Elementary Facts about Baire and Baire-Type Spaces
Abstract This chapter contains classical results about Baire-type conditions (Baire-like, b-Baire-like, CS-barrelled, s-barrelled) on tvs. We include applications to closed graph theorems and C(X) spaces. We also provide the first proof in book form of a remarkable result of Saxon (extending earlier results of Arias de Reyna and Valdivia), that states that, under Martin’s axiom, every lcs containing a dense hyperplane contains a dense non-Baire hyperplane. This part also contains analytic characterizations of certain completely regular Hausdorff spaces X. For example, we show that X is pseudocompact, is Warner bounded, or Cc(X) is a (df )-space if and only if for each sequence (μn)n in the dual Cc(X) there exists a sequence (tn)n ⊂ (0, 1] such that (tnμn)n is weakly bounded, strongly bounded, or equicon- tinuous, respectively. These characterizations help us produce a (df )-space Cc(X) that is not a (DF )-space, solving a basic and long-standing open question.
2.1 Baire spaces and Polish spaces
Let A be a subset of a nonvoid Hausdorff topological space X. We shall say that A is nowhere dense (or rare) if its closure A has a void interior. Clearly, every subset of a nowhere dense set is nowhere dense. A is called of first category if it is a countable union of nowhere dense subsets of X. Clearly, every subset of a first category set is again of first category. A is said to be of second category in X if it is not of first category. If A is of second category and A ⊂ B, then B is of second category. The classical Baire category theorem states the following.
Theorem 2.1 If E is either a complete metric or a locally compact Hausdorff space, then the intersection of countably many dense, open subsets of X is dense in E.
Proof We show only that the intersection of countably many dense open sets in every metric complete space (E, d) is nonvoid. If this were false, then we would = have E n En, where each En is a closed subset with empty interior. Hence there exists x1 and 0 <ε1 < 1 such that B(x1,ε1) ⊂ E \ E1, where B(x1,ε1) is the open −1 −1 ball at x1 with radius ε1. Next there exists x2 ∈ B(x1, 2 ε1) and 0 <ε2 < 2 ε1
J. Kakol ˛ et al., Descriptive Topology in Selected Topics of Functional Analysis, 13 Developments in Mathematics 24, DOI 10.1007/978-1-4614-0529-0_2, © Springer Science+Business Media, LLC 2011 14 2 Elementary Facts about Baire and Baire-Type Spaces such that B(x2,ε2) ⊂ E \E2. Continuing this way, one obtains a shrinking sequence −n of open balls B(xn,εn) with radius less than 2 disjoint with En. Clearly, (xn)n is a Cauchy sequence in (E, d), so it converges to x ∈ E \ En, n ∈ N, and we have reached a contradiction. Similarly, one gets that each open subset of (E, d) is of second category.
This deep theorem is a principal one in analysis and topology, providing many applications for the closed graph theorems and the uniform boundedness theorem. A topological space X is called a Baire space if every nonvoid open subset of X is of second category (equivalently, if the conclusion of the Baire theorem holds). Clearly, every Baire space is of second category. Although there exist topological spaces of second category that are not Baire spaces, we note that all tvs’s considered in the sequel are assumed to be real or complex if not specified otherwise.
Proposition 2.1 If a tvs E is of second category, E is a Baire space.
Proof Let A be a nonvoid open subset of E.Ifx ∈ A, then there exists a balanced + ⊂ = neighborhood of zero U in E such that x U A. Since E n nU and E is of second category, there exists m ∈ N such that mU is of second category. Then U is of second category, too. This implies that x + U is of second category and A, containing x + U, is also of second category.
We shall also need the following classical fact; see [328, 10.1.26]. For a com- pletely regular Hausdorff space X,byCc(X) and Cp(X) we denote the space of real-valued continuous functions on X endowed with the compact-open and point- wise topologies, respectively.
Proposition 2.2 Let X be a paracompact and locally compact topological space. Then Cc(X) is a Baire space.
Proof Since X is a paracompact locally compact space, X can be represented as the topological direct sum of a disjoint family {Xt : t ∈ T } of locally compact σ - compact spaces Xt , and we have a topological isomorphism of Cc(X) and the prod- uct t∈T Cc(Xt ). It is known that each space Cc(Xt ) is a Fréchet space (i.e., a metrizable and complete lcs) and since products of Fréchet spaces are Baire spaces (see Theorem 14.2 below for an alternative proof), we conclude that t∈T Cc(Xt ) is a Baire space.
The following general fact is a simple consequence of definitions above.
Proposition 2.3 If E is a tvs and F is a vector subspace of E, then F is either dense or nowhere dense in E. If F is dense in E and F is Baire, then E is Baire.
Proof Assume that F is not dense in E.LetG be its closure in E, a proper closed subspace of E.IfG is not nowhere dense in E, then there exists a bal- 2.1 Baire spaces and Polish spaces 15 anced neighborhood of zero U in E and a point x ∈ G such that x + U ⊂ G. Then = ⊂ E n nU G, providing a contradiction. The other part is clear.
Recall that every Cech-completeˇ space E (i.e., E can be represented as a countable intersection of open subsets of a compact space) is a Baire space. Arkhangel’skii proved [31] that if E is a topological group and F is a Cech-completeˇ subspace of E, then either F is nowhere dense in E or E is Cech-completeˇ as well. This, combined with Proposition 2.3, shows that if a tvs E contains a dense Cech-ˇ complete vector subspace, then E is Cech-complete.ˇ A subset A of a topological space X is said to have the Baire property in X if there exists an open subset U of X such that U \ A and A \ U are of first category. Let D(A) be the set of all x ∈ X such that each neighborhood U(x) of x inter- sects A in a set of second category. Set O(A) := int D(A).
Proposition 2.4 A subset A of a topological space X has the Baire property if and only if O(A)\ A is of first category.
Proof Assume A has the Baire property, and let U be an open set in X such that U \ A and A \ U are sets of first category. Note that D(A) ⊂ U. Indeed, if x ∈ D(A) \ U, then (by definition) the set
(X \ U)∩ A(= A \ U) is of second category. By the assumption, A \ U ⊂ A \ U is of first category, a contradiction. Since U \ U is nowhere dense, one concludes that U \ A is of first category. Finally, since
O(A)\ A ⊂ D(A) \ A ⊂ U \ A, then O(A)\ A is of first category as claimed. Now assume that O(A) \ A is a set of first category. It is enough to prove that A \ O(A) is of first category. Let C(A) be the union of the family L := {Ai : i ∈ I} of all the open subsets of X that intersect A in a set of first category. Note that O(A) = X \ C(A). We show that
A ∩ C(A) (= A \ O(A))
{ : ∈ } is of first category. Let Ai i J be a maximal pairwise disjoint subfamily of L. ∩ ∩ Then (as is easily seen) A ( i∈J Ai) is of first category. Then the set A i∈J Ai ⊂ is also of first category. By the maximality condition, we deduce that i∈I Ai i∈J Ai , which completes the proof.
Since A ∩ C(A) is of the first category, we note the following simple fact; see, for example, [421,p.4]. 16 2 Elementary Facts about Baire and Baire-Type Spaces
Proposition 2.5 Let E be a topological space, and let B be a subset of E that is \ { : ∈ N} the union of a sequence (Un)n of subsets of E. Then D(B) O(Un) n is nowhere dense. Therefore O(B)\ {O(Un) : n ∈ N} is also nowhere dense. Proof Assume that the interior A of the closed set D(B) \ {O(Un) : n ∈ N} is non-void. Then A ∩ B is of second category. Hence there exists m ∈ N such that A ∩ Um is of second category. Therefore, as Um ∩ C(Um) is of first category, we have A ⊂ C(Um), and hence A ∩ O(Um) is nonvoid, a contradiction.
Every Borel set in a topological space E has the Baire property. This easily fol- lows from the following well-known fact; see [371].
Proposition 2.6 Let E be a topological space. The family of all subsets of E with the Baire property forms a σ -algebra.
Now we are ready to formulate the following useful fact.
Proposition 2.7 Let U be a subset of a topological vector space E. If U is of second category and has the Baire property, U − U is a neighborhood of zero.
Proof Since U is of second category, we have O(U) = ∅.Ifx/∈ U −U, then clearly (x + U)∩ U =∅. The Baire property of U implies that (x + O(U)) ∩ O(U) is a set of first category. On the other hand, since nonvoid open subsets of O(U) are of second category, it follows that (x + O(U))∩ O(U) =∅. Then x/∈ O(U)− O(U). This proves that U − U contains the neighborhood of zero O(U)− O(U).
It is clear that the fact above has a corresponding variant for topological groups (called the Philips lemma); see, for example, [346]. In general, even the self-product X ×X of a Baire space X need not be Baire; see [324], [106], [164]. Nevertheless, the product i∈I Xi of metric complete spaces is { : ∈ } Baire; see [328]. Also, the product i∈I Xi of any family Xi i I of separable Baire spaces is a Baire space; see [421]. Arias de Reina [16] proved the following remarkable theorem.
2 Theorem 2.2 (Arias de Reina) The Hilbert space (ω1) contains a family {Xt : t<ω1} of different Baire subspaces such that for all t,u < ω1, t = u, the product Xt × Xu is not Baire.
Valdivia [424] generalized this result by proving the same conclusion in each p space c0(I) and (I), for uncountable set I , and 0
Proposition 2.8 (i) The intersection of any countable family of Polish subspaces of a topological space E is a Polish space. (ii) Every open (closed) subspace V of a Polish space E is a Polish space. Hence a subspace of a Polish space that is a Gδ-set is a Polish space. := Proof (i) Let (En) n be a sequence of Polish subspaces of E, and let G n En. Then the product n En (endowed with the product topology) and the diagonal ⊂ Δ n En (as a closed subset) are Polish spaces. Since Δ is homeomorphic to the intersection G, the conclusion follows. (ii) Let d be a complete metric on E.LetV be open and V c := E \ V . Define the function d(x,V c) by d(x,V c) := inf {d(x,y) : y ∈ V c}. Set ξ(x):= d(x,V c)−1 and D(x,y) := |ξ(x)− ξ(y)|+d(x,y) for all x,y ∈ V . It is easy to see that D(x,y) defines a complete metric on V giving the original topology of E restricted to V . Hence V is a Polish space.
The following characterizes Polish subspaces of a Polish space; see [371].
Proposition 2.9 A subspace F of a Polish space E is Polish if and only if F is a Gδ-set in E.
Proof If F is a Gδ-set in E, then F is a Polish space by the previous proposition. To prove the converse, let d (resp. d1) be a compatible (resp. complete) metric on −1 E (resp. F ). For each x ∈ F and each n ∈ N, there exists 0